Berger inequality

For a compact Riemannian manifold , let

where is the ball around with radius , be the injectivity radius, and set . Then the inequality

holds, with equality if and only if is isometric to the standard sphere with diameter .

This inequality relies on the Kazdan inequality applied to the Jacobi equation for operators on for a unit vector . Here, is the curvature operator, is the parallel transport along the geodesic ray , and is the parallel translated curvature operator on .

References

[a1]	M. Berger, "Une borne inférieure pour le volume d'une variété riemannienes en fonction du rayon d'injectivité" Ann. Inst. Fourier (Grenoble) , 30 (1980) pp. 259–265
[a2]	I. Chavel, "Riemannian geometry: A modern introduction" , Cambridge Univ. Press (1995)